Emergent neural dynamics and geometry for generalization in a transitive inference task.

Relational cognition-the ability to infer relationships that generalize to novel combinations of objects-is fundamental to human and animal intelligence. Despite this importance, it remains unclear how relational cognition is implemented in the brain due in part to a lack of hypotheses and predictions at the levels of collective neural activity and behavior. Here we discovered, analyzed, and experimentally tested neural networks (NNs) that perform transitive inference (TI), a classic relational task (if A > B and B > C, then A > C). We found NNs that (i) generalized perfectly, despite lacking overt transitive structure prior to training, (ii) generalized when the task required working memory (WM), a capacity thought to be essential to inference in the brain, (iii) emergently expressed behaviors long observed in living subjects, in addition to a novel order-dependent behavior, and (iv) expressed different task solutions yielding alternative behavioral and neural predictions. Further, in a large-scale experiment, we found that human subjects performing WM-based TI showed behavior inconsistent with a class of NNs that characteristically expressed an intuitive task solution. These findings provide neural insights into a classical relational ability, with wider implications for how the brain realizes relational cognition.

A mathematical theory of relational generalization in transitive inference.

Humans and animals routinely infer relations between different items or events and generalize these relations to novel combinations of items. This allows them to respond appropriately to radically novel circumstances and is fundamental to advanced cognition. However, how learning systems (including the brain) can implement the necessary inductive biases has been unclear. Here we investigated transitive inference (TI), a classic relational task paradigm in which subjects must learn a relation (A > B and B > C) and generalize it to new combinations of items (A > C). Through mathematical analysis, we found that a broad range of biologically relevant learning models (e.g. gradient flow or ridge regression) perform TI successfully and recapitulate signature behavioral patterns long observed in living subjects. First, we found that models with item-wise additive representations automatically encode transitive relations. Second, for more general representations, a single scalar "conjunctivity factor" determines model behavior on TI and, further, the principle of norm minimization (a standard statistical inductive bias) enables models with fixed, partly conjunctive representations to generalize transitively. Finally, neural networks in the "rich regime," which enables representation learning and has been found to improve generalization, unexpectedly show poor generalization and anomalous behavior. We find that such networks implement a form of norm minimization (over hidden weights) that yields a local encoding mechanism lacking transitivity. Our findings show how minimal statistical learning principles give rise to a classical relational inductive bias (transitivity), explain empirically observed behaviors, and establish a formal approach to understanding the neural basis of relational abstraction.

An active neural mechanism for relational learning and fast knowledge reassembly.

How do we gain general insights from limited novel experiences? Humans and animals have a striking ability to learn relationships between experienced items, enabling efficient generalization and rapid assimilation of new information. One fundamental instance of such relational learning is transitive inference (learn A>B and B>C, infer A>C), which can be quickly and globally reorganized upon learning a new item (learn A>B>C and D>E>F, then C>D, and infer B>E). Despite considerable study, neural mechanisms of transitive inference and fast reassembly of existing knowledge remain elusive. Here we adopt a meta-learning ("learning-to-learn") approach. We train artificial neural networks, endowed with synaptic plasticity and neuromodulation, to be able to learn novel orderings of arbitrary stimuli from repeated presentation of stimulus pairs. We then obtain a complete mechanistic understanding of this discovered neural learning algorithm. Remarkably, this learning involves active cognition: items from previous trials are selectively reinstated in working memory, enabling delayed, self-generated learning and knowledge reassembly. These findings identify a new mechanism for relational learning and insight, suggest new interpretations of neural activity in cognitive tasks, and highlight a novel approach to discovering neural mechanisms capable of supporting cognitive behaviors.

Semiclassical treatment of Feshbach resonances by transfer matrices.

A semiclassical method is presented for the calculation of Feshbach resonance positions and widths. This approach, based on semiclassical transfer matrices, relies only on relatively short trajectory fragments, thus avoiding problems associated with the long trajectories needed in more straightforward semiclassical techniques. Complex resonance energies are obtained from an implicit equation that is developed to compensate for the inaccuracy of the stationary phase approximation underlying the semiclassical transfer matrix applications. Although this treatment requires calculation of transfer matrices for complex energies, an initial value representation method makes it possible to extract such quantities from ordinary real-valued classical trajectories. This treatment is applied to obtain positions and widths for resonances in a model two-dimensional system, and the results are compared to those obtained from accurate quantum mechanical calculations. The semiclassical method successfully captures the irregular energy dependence of resonance widths that vary over a range of more than two orders of magnitude. An explicit semiclassical expression for the width of narrow resonances is also presented and serves as a simpler, useful approximation for many cases.

Imagination as a fundamental function of the hippocampus.

Imagination is a biological function that is vital to human experience and advanced cognition. Despite this importance, it remains unknown how imagination is realized in the brain. Substantial research focusing on the hippocampus, a brain structure traditionally linked to memory, indicates that firing patterns in spatially tuned neurons can represent previous and upcoming paths in space. This work has generally been interpreted under standard views that the hippocampus implements cognitive abilities primarily related to actual experience, whether in the past (e.g. recollection, consolidation), present (e.g. spatial mapping) or future (e.g. planning). However, relatively recent findings in rodents identify robust patterns of hippocampal firing corresponding to a variety of alternatives to actual experience, in many cases without overt reference to the past, present or future. Given these findings, and others on hippocampal contributions to human imagination, we suggest that a fundamental function of the hippocampus is to generate a wealth of hypothetical experiences and thoughts. Under this view, traditional accounts of hippocampal function in episodic memory and spatial navigation can be understood as particular applications of a more general system for imagination. This view also suggests that the hippocampus contributes to a wider range of cognitive abilities than previously thought. This article is part of the theme issue 'Thinking about possibilities: mechanisms, ontogeny, functions and phylogeny'.

Constant Sub-second Cycling between Representations of Possible Futures in the Hippocampus.

Cognitive faculties such as imagination, planning, and decision-making entail the ability to represent hypothetical experience. Crucially, animal behavior in natural settings implies that the brain can represent hypothetical future experience not only quickly but also constantly over time, as external events continually unfold. To determine how this is possible, we recorded neural activity in the hippocampus of rats navigating a maze with multiple spatial paths. We found neural activity encoding two possible future scenarios (two upcoming maze paths) in constant alternation at 8 Hz: one scenario per ∼125-ms cycle. Further, we found that the underlying dynamics of cycling (both inter- and intra-cycle dynamics) generalized across qualitatively different representational correlates (location and direction). Notably, cycling occurred across moving behaviors, including during running. These findings identify a general dynamic process capable of quickly and continually representing hypothetical experience, including that of multiple possible futures.

Three brain states in the hippocampus and cortex.

Contemporary brain research seeks to understand how cognition is reducible to neural activity. Crucially, much of this effort is guided by a scientific paradigm that views neural activity as essentially driven by external stimuli. In contrast, recent perspectives argue that this paradigm is by itself inadequate and that understanding patterns of activity intrinsic to the brain is needed to explain cognition. Yet, despite this critique, the stimulus-driven paradigm still dominates-possibly because a convincing alternative has not been clear. Here, we review a series of findings suggesting such an alternative. These findings indicate that neural activity in the hippocampus occurs in one of three brain states that have radically different anatomical, physiological, representational, and behavioral correlates, together implying different functional roles in cognition. This three-state framework also indicates that neural representations in the hippocampus follow a surprising pattern of organization at the timescale of ∼1 s or longer. Lastly, beyond the hippocampus, recent breakthroughs indicate three parallel states in the cortex, suggesting shared principles and brain-wide organization of intrinsic neural activity.

Semiclassical tunneling splittings for arbitrary vibrational states in multidimensional double wells.

A semiclassical theory developed in a previous paper [K. G. Kay, Phys. Rev. A 96, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.

Distinct hippocampal-cortical memory representations for experiences associated with movement versus immobility.

While ongoing experience proceeds continuously, memories of past experience are often recalled as episodes with defined beginnings and ends. The neural mechanisms that lead to the formation of discrete episodes from the stream of neural activity patterns representing ongoing experience are unknown. To investigate these mechanisms, we recorded neural activity in the rat hippocampus and prefrontal cortex, structures critical for memory processes. We show that during spatial navigation, hippocampal CA1 place cells maintain a continuous spatial representation across different states of motion (movement and immobility). In contrast, during sharp-wave ripples (SWRs), when representations of experience are transiently reactivated from memory, movement- and immobility-associated activity patterns are most often reactivated separately. Concurrently, distinct hippocampal reactivations of movement- or immobility-associated representations are accompanied by distinct modulation patterns in prefrontal cortex. These findings demonstrate a continuous representation of ongoing experience can be separated into independently reactivated memory representations.

Applying Bogomolny's quantization method to generic classical systems.

The quantization method of Bogomolny [Nonlinearity 5, 805 (1992)] can potentially provide semiclassical estimates for energy levels of all bound states of arbitrary systems. This approach requires the formation of the transfer matrix TE as a function of energy E. Existing practical methods for calculating this matrix require a recalculation of many classical trajectories for each energy. This has hampered the application of Bogomolny's method to generic systems that do not possess special classical scaling properties. Generalizing earlier work [H. Barak and K. G. Kay, Phys. Rev. E 88, 062926 (2013)], we develop initial value representation formulas for TE that overcome this problem. These expressions are obtained from a generalized Herman-Kluk formula for the propagator that allows one to easily derive a family of semiclassical integral approximations for the Green's function that are, in turn, used to form the transfer matrix. Calculations for two-dimensional systems show that Bogomolny's method with the present expressions for TE produces accurate semiclassical energy levels from small transfer matrices.

Apolipoprotein E4 Causes Age-Dependent Disruption of Slow Gamma Oscillations during Hippocampal Sharp-Wave Ripples.

Apolipoprotein (apo) E4 is the major genetic risk factor for Alzheimer's disease (AD), but the mechanism by which it causes cognitive decline is unclear. In knockin (KI) mice, human apoE4 causes age-dependent learning and memory impairments and degeneration of GABAergic interneurons in the hippocampal dentate gyrus. Here we report two functional apoE4-KI phenotypes involving sharp-wave ripples (SWRs), hippocampal network events critical for memory processes. Aged apoE4-KI mice had fewer SWRs than apoE3-KI mice and significantly reduced slow gamma activity during SWRs. Elimination of apoE4 in GABAergic interneurons, which prevents learning and memory impairments, rescued SWR-associated slow gamma activity but not SWR abundance in aged mice. SWR abundance was reduced similarly in young and aged apoE4-KI mice; however, the full SWR-associated slow gamma deficit emerged only in aged apoE4-KI mice. These results suggest that progressive decline of interneuron-enabled slow gamma activity during SWRs critically contributes to apoE4-mediated learning and memory impairments. VIDEO ABSTRACT.

A hippocampal network for spatial coding during immobility and sleep.

How does an animal know where it is when it stops moving? Hippocampal place cells fire at discrete locations as subjects traverse space, thereby providing an explicit neural code for current location during locomotion. In contrast, during awake immobility, the hippocampus is thought to be dominated by neural firing representing past and possible future experience. The question of whether and how the hippocampus constructs a representation of current location in the absence of locomotion has been unresolved. Here we report that a distinct population of hippocampal neurons, located in the CA2 subregion, signals current location during immobility, and does so in association with a previously unidentified hippocampus-wide network pattern. In addition, signalling of location persists into brief periods of desynchronization prevalent in slow-wave sleep. The hippocampus thus generates a distinct representation of current location during immobility, pointing to mnemonic processing specific to experience occurring in the absence of locomotion.

Wave packet propagation across barriers by semiclassical initial value methods.

Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.

Clusterless Decoding of Position from Multiunit Activity Using a Marked Point Process Filter.

Point process filters have been applied successfully to decode neural signals and track neural dynamics. Traditionally these methods assume that multiunit spiking activity has already been correctly spike-sorted. As a result, these methods are not appropriate for situations where sorting cannot be performed with high precision, such as real-time decoding for brain-computer interfaces. Because the unsupervised spike-sorting problem remains unsolved, we took an alternative approach that takes advantage of recent insights into clusterless decoding. Here we present a new point process decoding algorithm that does not require multiunit signals to be sorted into individual units. We use the theory of marked point processes to construct a function that characterizes the relationship between a covariate of interest (in this case, the location of a rat on a track) and features of the spike waveforms. In our example, we use tetrode recordings, and the marks represent a four-dimensional vector of the maximum amplitudes of the spike waveform on each of the four electrodes. In general, the marks may represent any features of the spike waveform. We then use Bayes's rule to estimate spatial location from hippocampal neural activity. We validate our approach with a simulation study and experimental data recorded in the hippocampus of a rat moving through a linear environment. Our decoding algorithm accurately reconstructs the rat's position from unsorted multiunit spiking activity. We then compare the quality of our decoding algorithm to that of a traditional spike-sorting and decoding algorithm. Our analyses show that the proposed decoding algorithm performs equivalent to or better than algorithms based on sorted single-unit activity. These results provide a path toward accurate real-time decoding of spiking patterns that could be used to carry out content-specific manipulations of population activity in hippocampus or elsewhere in the brain.

Initial-value representation of the semiclassical zeta function.

This work casts the semiclassical zeta function in a form suitable for practical calculations of energy levels for rather general systems. To accomplish this, the zeta function is approximated by applying an initial-value representation (IVR) treatment to the traces of the transfer matrix that appear when the function is expanded in cumulants. Because this approach does not require searches for periodic orbits or special trajectories obeying double-ended boundary conditions, it is easily applicable to multidimensional systems with smooth potentials. Calculations are presented for the energy levels of three two-dimensional systems, including one that is classically integrable, one having mixed phase space, and one that is almost fully chaotic. The results show that the present treatment is far more numerically efficient than a previously proposed IVR method for the zeta function [Barak and Kay, Phys. Rev. E 88, 062926 (2013)]. The approach described here successfully resolves nearly all energy levels in the range investigated for the first two systems as well as energy levels in spectral regions that are not too highly congested for the highly chaotic system.

Complex time paths for semiclassical wave packet propagation with complex trajectories.

The use of complex-valued trajectories in semiclassical wave packet methods can lead to problems that prevent calculation of the wave function in certain regions of the configuration space. We investigate this so-called bald spot problem in the context of generalized Gaussian wave packet dynamics. The analysis shows that the bald spot phenomenon is essentially due to the complex nature of the initial conditions for the trajectories. It is, therefore, expected to be a general feature of several semiclassical methods that rely on trajectories with such initial conditions. A bald region is created when a trajectory, needed to calculate the wave function at a given time, reaches a singularity of the potential energy function in the complex plane at an earlier, real time. This corresponds to passage of a branch point singularity across the real axis of the complex time plane. The missing portions of the wave function can be obtained by deforming the time path for the integration of the equations of motion into the complex plane so that the singularity is circumvented. We present examples of bald spots, singularity times, and suitable complex time paths for one-dimensional barrier transmission in the Eckart and Gaussian systems. Although the bald regions for the Eckart system are often localized, they are found to be semi-infinite for the Gaussian system. For the case of deep tunneling, the bald regions for both systems may encompass the entire portion of space occupied by the transmitted wave packet. Thus, the use of complex time paths becomes essential for a treatment of barrier tunneling.

Evaluating the Accuracy of Hessian Approximations for Direct Dynamics Simulations.

Direct dynamics simulations are a very useful and general approach for studying the atomistic properties of complex chemical systems, since an electronic structure theory representation of a system's potential energy surface is possible without the need for fitting an analytic potential energy function. In this paper, recently introduced compact finite difference (CFD) schemes for approximating the Hessian [J. Chem. Phys.2010, 133, 074101] are tested by employing the monodromy matrix equations of motion. Several systems, including carbon dioxide and benzene, are simulated, using both analytic potential energy surfaces and on-the-fly direct dynamics. The results show, depending on the molecular system, that electronic structure theory Hessian direct dynamics can be accelerated up to 2 orders of magnitude. The CFD approximation is found to be robust enough to deal with chaotic motion, concomitant with floppy and stiff mode dynamics, Fermi resonances, and other kinds of molecular couplings. Finally, the CFD approximations allow parametrical tuning of different CFD parameters to attain the best possible accuracy for different molecular systems. Thus, a direct dynamics simulation requiring the Hessian at every integration step may be replaced with an approximate Hessian updating by tuning the appropriate accuracy.

Semiclassical initial-value representation of the transfer operator.

The ability of semiclassical initial-value representation (IVR) methods to determine approximate energy levels for bound systems is limited due to problems associated with long classical trajectories. These difficulties become especially severe for large or classically chaotic systems. This work attempts to overcome such problems by developing an IVR expression that is classically equivalent to Bogomolny's formula for the transfer matrix [E. B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)] and can be used to determine semiclassical energy levels. The method is adapted to levels associated with states of desired symmetries and applied to two two-dimensional quartic oscillator systems, one integrable and one mostly chaotic. For both cases, the technique is found to resolve all energy levels in the ranges investigated. The IVR method does not require a search for special trajectories obeying boundary conditions on the Poincaré surface of section and leads to more rapid convergence of Monte Carlo phase space integrations than a previously developed IVR technique. It is found that semiclassical energies can be extracted from the eigenvalues of transfer matrices of dimension close to the theoretical minimum determined by Bogomolny's theory. The results support the assertion that the present IVR theory provides a different semiclassical approximation to the transfer matrix than that of Bogomolny for ℏ≠0. For the chaotic system investigated the IVR energies are found to be generally more accurate than those predicted by Bogomolny's theory.

Semiclassical initial value approximation for Green's function.

A semiclassical initial value approximation is obtained for the energy-dependent Green's function. For a system with f degrees of freedom the Green's function expression has the form of a (2f-1)-dimensional integral over points on the energy surface and an integral over time along classical trajectories initiated from these points. This approximation is derived by requiring an integral ansatz for Green's function to reduce to Gutzwiller's semiclassical formula when the integrations are performed by the stationary phase method. A simpler approximation is also derived involving only an (f-1)-dimensional integral over momentum variables on a Poincare surface and an integral over time. The relationship between the present expressions and an earlier initial value approximation for energy eigenfunctions is explored. Numerical tests for two-dimensional systems indicate that good accuracy can be obtained from the initial value Green's function for calculations of autocorrelation spectra and time-independent wave functions. The relative advantages of initial value approximations for the energy-dependent Green's function and the time-dependent propagator are discussed.

Regulation of parkinsonian motor behaviours by optogenetic control of basal ganglia circuitry.

Neural circuits of the basal ganglia are critical for motor planning and action selection. Two parallel basal ganglia pathways have been described, and have been proposed to exert opposing influences on motor function. According to this classical model, activation of the 'direct' pathway facilitates movement and activation of the 'indirect' pathway inhibits movement. However, more recent anatomical and functional evidence has called into question the validity of this hypothesis. Because this model has never been empirically tested, the specific function of these circuits in behaving animals remains unknown. Here we report direct activation of basal ganglia circuitry in vivo, using optogenetic control of direct- and indirect-pathway medium spiny projection neurons (MSNs), achieved through Cre-dependent viral expression of channelrhodopsin-2 in the striatum of bacterial artificial chromosome transgenic mice expressing Cre recombinase under control of regulatory elements for the dopamine D1 or D2 receptor. Bilateral excitation of indirect-pathway MSNs elicited a parkinsonian state, distinguished by increased freezing, bradykinesia and decreased locomotor initiations. In contrast, activation of direct-pathway MSNs reduced freezing and increased locomotion. In a mouse model of Parkinson's disease, direct-pathway activation completely rescued deficits in freezing, bradykinesia and locomotor initiation. Taken together, our findings establish a critical role for basal ganglia circuitry in the bidirectional regulation of motor behaviour and indicate that modulation of direct-pathway circuitry may represent an effective therapeutic strategy for ameliorating parkinsonian motor deficits.